Mr. Smith was murdered in Kansas City. The police determined that the time of death was
between 11:10 pm and 11:30 pm. Four suspects were questioned: Butler, the butler; Cook,
the cook; Ruby, the maid; and Irma, Mr. Smith’s secretary.

The suspects made the
following statements:
Butler: I did not do it. Irma did it. Mr. Smith was blackmailing Irma. Ruby and I were
watching television together from 10:10 p.m. until 12:30 a.m.
Cook: I am innocent. Irma was being blackmailed. Butler murdered Mr. Smith. I saw
Irma leave the house at 10:00 p.m.
Ruby: I am innocent. Butler and I were watching television together at the time of the
murder. Irma was being blackmailed. I saw Irma speaking to Mr. Smith at 9:30
p.m. on the night of the murder.
Irma: I did not kill Mr. Smith. I was not being blackmailed. I was in St. Louis during
the entire night of the murder. Ruby is the murderess.

If each suspect made exactly two true statements and told exactly two lies, determine who
killed Mr. Smith.

It's true, it is an easy whodunit. Nevertheless it is an amusing and interesting one: There is a need to distuinguish conditional from biconditional connections. For example, from the truth of the statements by the butler and Ruby, saying that they were watching tv together at the time of the murder, follows the truth of their innocence. But from the truth of their innocence does not follow that they were watching tv at the time of the murder. And from the falsity of what they are saying, it does not follow that either the butler or Ruby is the murderer.

The proof below makes only sense if one is interested to see as many small steps as possible, otherwise it is ridiculous. If one is interested to see an efficient method of reasoning, we can look at the far better proof given by hoodat.

We know that only one of the four suspects (B,C,R,I) did the murder (premises 1 - 7). Our goals are to show that neither B nor C nor I can be the murderer (three subproofs) and therefore R must be the murderess.

The proof begins with 55 premises. Many of them are superflous (especially the doubles no. 30 and no. 46).

Propositional variables:

B = Butler is the murderer

B1-B4 = Each for the true statements of Butler

C = Cook is the murderer

C1-C4 = Each for the true statements of Cook

R = Ruby is the murderess

R1-R4 = Each for the true statements of Ruby

I = Irma is the murderess

I1-I4 = Each for the true statements of Irma

Premises 1 - 7:

The murderer was one of the four suspects and acted alone:

1. B v C v R v I

2. ~(B & R)

3. ~(B & C)

4. ~(B & I)

5. ~(R & C)

6. ~(R & I)

7. ~(C & I)

Premises 8 - 11:

Each suspect made two true statements and told two lies:

8. ( ((B1 & B2) & ~(B3 v B4)) v ((B1 & B3) & ~(B2 v B4))

v ((B1 & B4) & ~(B2 v B3)) v ((B3 & B4) & ~(B1 v B2))

v ((B2 & B4) & ~(B1 v B3)) v ((B2 & B3) & ~(B1 v B4)) )

9. ( ((R1 & R2) & ~(R3 v R4)) v ((R1 & R3) & ~(R2 v R4))

v ((R1 & R4) & ~(R2 v R3)) v ((R3 & R4) & ~(R1 v R2))

v ((R2 & R4) & ~(R1 v R3)) v ((R2 & R3) & ~(R1 v R4)) )

10. ( ((C1 & C2) & ~(C3 v C4)) v ((C1 & C3) & ~(C2 v C4))

v ((C1 & C4) & ~(C2 v C3)) v ((C3 & C4) & ~(C1 v C2))

v ((C2 & C4) & ~(C1 v C3)) v ((C2 & C3) & ~(C1 v C4)) )

11. ( ((I1 & I2) & ~(I3 v I4)) v ((I1 & I3) & ~(I2 v I4))

v ((I1 & I4) & ~(I2 v I3)) v ((I3 & I4) & ~(I1 v I2))

v ((I2 & I4) & ~(I1 v I3)) v ((I2 & I3) & ~(I1 v I4)) )

Premises 12 - 55:

Conditional or biconditional connections from the suspects' statements: